5: Probability Concepts

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Chapter 5 Probability Concepts
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In Chapter 5

• 5.1 What is Probability?
• 5.2 Types of Random Variables
• 5.3 Discrete Random Variables
• 5.4 Continuous Random Variables
• 5.5 More Rules and Properties of Probability

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Definitions

• Random variable a numerical quantity that takes
  on different values depending on chance
• Population the set of all possible values for a
  random variable
• Event an outcome or set of outcomes
• Probability the relative frequency of an event
  in the population alternatively the proportion
  of times an event is expected to occur in the
  long run

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Example

• In a given year 42,636 traffic fatalities
  (events) in a population of N 293,655,000
• Random sample population
• Probability of event relative freq in pop
  42,636 / 293,655,000 .0001452 1 in 6887

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Example Probability

• Assume, 20 of population has a condition
• Repeatedly sample population
• The proportion of observations positive for the
  condition approaches 0.2 after a very large
  number of trials

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Quantifying Uncertainty
Probability Expression
0.00 Never
0.05 Seldom
0.20 Infrequent
0.50 As often as not
0.80 Very frequent
0.95 Highly likely
1.00 Always
Probability is used to quantify levels of belief
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Random Variables

• Random variable a numerical quantity that takes
  on different values depending on chance
• Two types of random variables
• Discrete random variables (countable set of
  possible outcomes)
• Continuous random variable (unbroken chain of
  possible outcomes)

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Example Discrete Random Variable

• Treat 4 patients with a drug that is 75
  effective
• Let X the variable number of patients that
  respond to treatment
• X is a discrete random variable can be either 0,
  1, 2, 3, or 4 (a countable set of possible
  outcomes)

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Example Discrete Random Variable

• Discrete random variables are understood in terms
  of their probability mass function (pmf)
• pmf a mathematical function that assigns
  probabilities to all possible outcomes for a
  discrete random variable.
• This table shows the pmf for our four patients
  example

x 0 1 2 3 4
Pr(Xx) 0.0039 0.0469 0.2109 0.4219 0.3164
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The four patients pmf can also be shown
graphically
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Area on pmf Probability
Four patients pmf

• Areas under pmf graphs correspond to probability
• For example Pr(X 2) shaded rectangle
  height base .2109 1.0 .2109

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Example Continuous Random Variable

• Continuous random variables have an infinite set
  of possible outcomes
• Example generate random numbers with this
  spinner ?
• Outcomes form a continuum between 0 and 1

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Example Continuous Random Variable

• probability density function (pdf) a
  mathematical function that assigns probabilities
  for continuous random variables
• The probability of any exact value is 0
• BUT, the probability of a range is the area under
  the pdf curve (bottom)

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Example Continuous Random Variable

• Area probabilities
• The pdf for the random spinner variable ?
• The probability of a value between 0 and 0.5 Pr(0
  X 0.5) shaded rectangle height base
  1 0.5 0.5

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pdfs come in various shapeshere are examples
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Areas Under the Curve

• pdf curves are analogous to probability
  histograms
• AREAS probabilities
• Top figure histogram, ages 9 shaded
• Bottom figure pdf, ages 9 shaded
• Both represent proportion of population 9

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Properties of Probabilities

• Property 1. Probabilities are always between 0
  and 1
• Property 2. The sample space (S) for a random
  variable represents all possible outcomes and
  must sum to 1 exactly.
• Property 3. The probability of the complement of
  an event (NOT the event) 1 MINUS the
  probability of the event.
• Property 4. Probabilities of disjoint events can
  be added.

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Properties of Probabilities In symbols

• Property 1. 0 Pr(A) 1
• Property 2. Pr(S) 1
• Property 3. Pr(A) 1 Pr(A), A represents the
  complement of A
• Property 4. Pr(A or B) Pr(A) Pr(B) when A
  and B are disjoint

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Properties 1 2 Illustrated

• Property 1. Note that all probabilities are
  between 0 and 1.
• Property 2. The sample space sums to 1Pr(S)
  .0039 .0469 .2109 .4219 .3164 1

Four patients pmf
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Property 3 (Complements)

• Let A 4 successes
• Then, A not A 3 or fewer successes
• Property of complements
• Pr(A) 1 Pr(A) 1 0.3164 0.6836

Four patients pmf
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Property 4 (Disjoint Events)

• Let A represent 4 successes
• Let B represent 3 successes
• A B are disjoint
• The probability of observing 3 or 4Pr(A or B)
• Pr(A) Pr(B)
• 0.3164 0.4129
• 0.7293

Four patients pmf
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Cumulative ProbabilityLeft tail

• Cumulative probability probability of x or less
  
• Denoted Pr(X x)
• Corresponds to area in left tail
• Example Pr(X 2) area in left tail .0039
  .0469 .2109 0.2617

.2109
.0469
.0039
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Right tail

• Probabilities greater than a value are denoted
  Pr(X gt x)
• Complement of cumulative probability
• Corresponds to area in right tail of distribution
• Example (4 patients pmf) Pr (X gt 3)
  complement of Pr(X 2) 1 - 0.2617 .7389

.2109
.0469
.0039
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Mean and Variance of a Discrete RV
Definitional formulas not covered in some courses.
The mean, i.e., expectation (see p. 95)
The variance (see p. 96)